The structure theory of Ricci limit spaces is established in [1]}, [2]}, [3]}. For example, the Laplacian \(\Delta \)  on \((X, \mathsf {d}, \mathfrak {m})\)  is well-defined via the rectifiablity as a metric measure space. Moreover it is proved in [3]} that the spectrums behave continuously with respect to the convergence (REF ) if \((X, \mathsf {d})\)  is compact, which confirms a conjecture raised in [5]}. This gives an analytic answer to the problem \((\star \star )\) .
